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arXiv · 1211.6291

Dyadic harmonic analysis beyond doubling measures

Luis Daniel López-Sánchez·José María Martell·Javier Parcet
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Abstract

We characterize the Borel measures $\mu$ on $\mathbb{R}$ for which the associated dyadic Hilbert transform, or its adjoint, is of weak-type $(1,1)$ and/or strong-type $(p,p)$ with respect to $\mu$. Surprisingly, the class of such measures is strictly bigger than the traditional class of dyadically doubling measures and strictly smaller than the whole Borel class. In higher dimensions, we provide a complete characterization of the weak-type $(1,1)$ for arbitrary Haar shift operators, cancellative or not, written in terms of two generalized Haar systems and these include the dyadic paraproducts. Our main tool is a new Calder\'on-Zygmund decomposition valid for arbitrary Borel measures which is of independent interest.

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Luis Daniel López-Sánchez, José María Martell, Javier Parcet. 2012-11-27. Dyadic harmonic analysis beyond doubling measures. https://doi.org/10.1016/j.aim.2014.08.001

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